Finite Math Examples

${(\sqrt[3]{27 x y^{5}})}^{2}$

Step 1

Rewrite $27 x y^{5}$ as ${(3 y)}^{3} (x y^{2})$ .

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Step 1.1

Rewrite $27$ as $3^{3}$ .

${\sqrt[3]{3^{3} x y^{5}}}^{2}$

Step 1.2

Factor out $y^{3}$ .

${\sqrt[3]{3^{3} x (y^{3} y^{2})}}^{2}$

Step 1.3

Move $x$ .

${\sqrt[3]{3^{3} y^{3} x y^{2}}}^{2}$

Step 1.4

Rewrite $3^{3} y^{3}$ as ${(3 y)}^{3}$ .

${\sqrt[3]{{(3 y)}^{3} x y^{2}}}^{2}$

Step 1.5

Add parentheses.

${\sqrt[3]{{(3 y)}^{3} (x y^{2})}}^{2}$

Step 2

Pull terms out from under the radical.

${(3 y \sqrt[3]{x y^{2}})}^{2}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $3 y \sqrt[3]{x y^{2}}$ .

${(3 y)}^{2} {\sqrt[3]{x y^{2}}}^{2}$

Step 3.2

Apply the product rule to $3 y$ .

$3^{2} y^{2} {\sqrt[3]{x y^{2}}}^{2}$

Step 4

Raise $3$ to the power of $2$ .

$9 y^{2} {\sqrt[3]{x y^{2}}}^{2}$

Step 5

Rewrite ${\sqrt[3]{x y^{2}}}^{2}$ as $\sqrt[3]{{(x y^{2})}^{2}}$ .

$9 y^{2} \sqrt[3]{{(x y^{2})}^{2}}$

Step 6

Apply the product rule to $x y^{2}$ .

$9 y^{2} \sqrt[3]{x^{2} {(y^{2})}^{2}}$

Step 7

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 7.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 y^{2} \sqrt[3]{x^{2} y^{2 \cdot 2}}$

Step 7.2

Multiply $2$ by $2$ .

$9 y^{2} \sqrt[3]{x^{2} y^{4}}$

Step 8

Rewrite $x^{2} y^{4}$ as $y^{3} (x^{2} y)$ .

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Step 8.1

Factor out $y^{3}$ .

$9 y^{2} \sqrt[3]{x^{2} (y^{3} y)}$

Step 8.2

Reorder $x^{2}$ and $y^{3}$ .

$9 y^{2} \sqrt[3]{y^{3} x^{2} y}$

Step 8.3

Add parentheses.

$9 y^{2} \sqrt[3]{y^{3} (x^{2} y)}$

Step 9

Pull terms out from under the radical.

$9 y^{2} (y \sqrt[3]{x^{2} y})$

Step 10

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 10.1

Move $y$ .

$9 (y \cdot y^{2}) \sqrt[3]{x^{2} y}$

Step 10.2

Multiply $y$ by $y^{2}$ .

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Step 10.2.1

Raise $y$ to the power of $1$ .

$9 (y^{1} y^{2}) \sqrt[3]{x^{2} y}$

Step 10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$9 y^{1 + 2} \sqrt[3]{x^{2} y}$

Step 10.3

Add $1$ and $2$ .

$9 y^{3} \sqrt[3]{x^{2} y}$